Topological gauge theories and group cohomology

Abstract

We show that three dimensional Chern-Simons gauge theories with a compact gauge groupG (not necessarily connected or simply connected) can be classified by the integer cohomology groupH 4(BG,Z). In a similar way, possible Wess-Zumino interactions of such a groupG are classified byH 3(G,Z). The relation between three dimensional Chern-Simons gauge theory and two dimensional sigma models involves a certain natural map fromH 4(BG,Z) toH 3(G,Z). We generalize this correspondence to topological “spin” theories, which are defined on three manifolds with spin structure, and are related to what might be calledZ 2 graded chiral algebras (or chiral superalgebras) in two dimensions. Finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group, establishing links with two dimensional (holomorphic) orbifold models.

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References

  1. 1.

    Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys.121, 351 (1989)

    Article  Google Scholar 

  2. 2.

    Moore, G., Seiberg, N.: Taming the conformal zoo. Phys. Lett.220B, 422 (1989)

    Google Scholar 

  3. 3.

    Chern, S.-S., Simons, J.: Characteristic forms and geometric invariants. Ann. Math.99, 48–69 (1974)

    Google Scholar 

  4. 4.

    Cheeger, J., Simons, J.: Differential characters and geometric invariants. In: Geometry and topology. Lecture Notes in Mathematics vol.1167. Berlin, Heidelberg, New York: Springer 1985

    Google Scholar 

  5. 5.

    Witten, E.: Non-Abelian Bosonization in two dimensions. Commun. Math. Phys.92, 455 (1984)

    Article  Google Scholar 

  6. 6.

    Segal, G.: Lecture at the IAMP Congress (Swansea, July 1988), and Oxford University preprint (to appear)

  7. 7.

    Dijkgraaf, R., Vafa, C., Verlinde, E., Verlinde, H.: Operator algebra of orbifold models. Commun. Math. Phys.123, 485 (1989)

    Article  Google Scholar 

  8. 8.

    Borel, A.: Topology of Lie groups and characteristic classes. Bull. A.M.S.61, 397–432, (1955)

    Google Scholar 

  9. 9.

    Borel, A., Hirzebruch, F.: Characteristic classes and homogeneous spaces I. Am. J. Math.80, 458–538 (1958)

    Google Scholar 

  10. 10.

    Milnor, J., Stasheff, J.: Characteristic classes. Annals of Mathematics Studies vol.76. Princeton, NJ: Princeton University Press 1974

    Google Scholar 

  11. 11.

    Madsen, I., Milgram, R.J.: The classifying spaces for surgery and cobordism of manifolds. Annals of Mathematics Studies vol.92. Princeton, NJ: Princeton University Press 1979

    Google Scholar 

  12. 12.

    Brown, K.S.: Cohomology of groups. Graduate Texts in Mathematics vol.87. Berlin, Heidelberg, New York: Springer 1982

    Google Scholar 

  13. 13.

    Freed, D.S.: Determinants, torsion, and strings. Commun. Math. Phys.107,483–514 (1986)

    Article  Google Scholar 

  14. 14.

    Milnor, J.: Construction of universal bundles II. Ann. Math.63, 430–436 (1956)

    Google Scholar 

  15. 15.

    Stong, R.E.: Notes on cobordism theory. Mathematical Notes. Princeton, NJ: Princeton University Press 1968

    Google Scholar 

  16. 16.

    Conner, P.E., Floyd, E.E.: Differentiable periodic maps. Bull. Am. Math. Soc.68, 76–86 (1962)

    Google Scholar 

  17. 17.

    Narasinhan, H.S., Ramanan, S.: Existence of universal connections. Am. J. Math.83, 563–572 (1961);85, 223–231 (1963)

    Google Scholar 

  18. 18.

    Felder, G., Gawedzki, K., Kupiainen, A.: Spectra of Wess-Zumino-Witten models with arbitrary simple groups. Commun. Math. Phys.117, 127–158 (1988)

    Article  Google Scholar 

  19. 19.

    Gepner, D., Witten, E.: String theory on group manifolds. Nucl. Phys.B278, 493–549 (1986)

    Article  Google Scholar 

  20. 20.

    Elitzur, S., Moore, G., Schwimmer, A., Seiberg, N.: Remarks on the canonical quantization of the Chern-Simons-Witten theory. Preprint IASSNS-HEP-89/20

  21. 21.

    Freed, D.S., Uhlenbeck, K.K.: Instantons and four-manifolds. Math. Sci. Res. Inst. Publ. vol.1, Berlin, Heidelberg, New York: Springer 1984

    Google Scholar 

  22. 22.

    't Hooft, G.: Some twisted self-dual solutions for the Yang-Mills equations on a hypertorus. Commun. Math. Phys.81, 167–275 (1981); van Baal, P.: Some results forSU(N) gauge fields on the hypertorus. Commun. Math. Phys.85, 529 (1982)

    Google Scholar 

  23. 23.

    Schellekens, A.N., Yankielowicz, S.: Extended Chiral algebras and modular invariant partition functions. Preprint CERN-TH5344/89

  24. 24.

    Karpilovsky, G.: Projective representations of finite groups. New York: Marcel Dekker 1985

    Google Scholar 

  25. 25.

    Verlinde, E.: Fusion rules and modular transformations in 2d conformal field theory. Nucl. Phys. B300, 360 (1988)

    Article  Google Scholar 

  26. 26.

    Hempel, J.: 3-Manifolds. Annals of Mathematics Studies vol.86. Princeton, NJ: Princeton University Press 1976

    Google Scholar 

  27. 27.

    Vafa, C.: Modular invariance and discrete torsion on orbifolds. Nucl. Phys. B273, 592 (1986)

    Article  Google Scholar 

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Communicated by A. Jaffe

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Dijkgraaf, R., Witten, E. Topological gauge theories and group cohomology. Commun.Math. Phys. 129, 393–429 (1990). https://doi.org/10.1007/BF02096988

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Keywords

  • Neural Network
  • Manifold
  • Statistical Physic
  • Complex System
  • Gauge Theory